I was wondering about the implications of the gauge/gravity (or AdS/CFT in a more restrictive sense) duality for the way we deal with physical theories, and I was wondering if the duality was believed to be exact, i.e. that the gauge theory and the gravity counterpart are describing exactly the same situation?
If the answer is yes, then as a follow-up question, does that mean that we would not be able to say how many dimensions reality has, in case we were to find an unified gauge theory in $d$ dimensions dual with a gravity theory in $d+1$ dimensions? We would also then not be able to differentiate if an effect is due to gravity or to gauge theory (for example in the case we were to find a gravity dual to QCD, although this is not a CFT)?
Or is it that the extra degree of freedom induced by the $+1$ dimension on the gravity side is somehow encoded in the gauge group?
Thank you in advance for your answers.
Edit: Funnily enough, I just came across this statement in the lecture notes by Năstase, which I guess answers the first question (the 2nd one remains):